Question: The matrix
\[\begin{pmatrix} a & \frac{15}{34} \\ c & \frac{25}{34} \end{pmatrix}\]corresponds to a projection.  Enter the ordered pair $(a,c).$
Solution: Suppose $\mathbf{P}$ is the matrix for projecting onto the vector $\mathbf{p}.$  Then for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is a scalar multiple of $\mathbf{p}.$  So when we apply the projection again to $\mathbf{P} \mathbf{v},$ the result is still $\mathbf{P} \mathbf{v}.$  This means
\[\mathbf{P} (\mathbf{P} \mathbf{v}) = \mathbf{P} \mathbf{v}.\]In other words, $\mathbf{P}^2 \mathbf{v} = \mathbf{P} \mathbf{v}.$  Since this holds for all vectors $\mathbf{v},$
\[\mathbf{P}^2 = \mathbf{P}.\]Here,
\[\mathbf{P}^2 = \begin{pmatrix} a & \frac{15}{34} \\ c & \frac{25}{34} \end{pmatrix} \begin{pmatrix} a & \frac{15}{34} \\ c & \frac{25}{34} \end{pmatrix} = \begin{pmatrix} a^2 + \frac{15}{34} c & \frac{15}{34} a + \frac{375}{1156} \\ ac + \frac{25}{34} c & \frac{15}{34} c + \frac{625}{1156} \end{pmatrix}.\]Thus, $\frac{15}{34} a + \frac{375}{1156} = \frac{15}{34}$ and $\frac{15}{34} c + \frac{625}{1156} = \frac{25}{34}.$  Solving, we find $(a,c) = \boxed{\left( \frac{9}{34}, \frac{15}{34} \right)}.$